John Stillwell

The main theme of the book under review is that new mathematical ideas develop as a result of trying to reconcile what seems at first sight impossible or what is actually impossible in some universe of discourse. Stillwell elaborates on this with chapters on irrational and imaginary numbers, parallel lines, infinitesimals, curved space, the fourth dimension, ideal numbers, periodic space, and the infinite. Even though these topics (except for ideal numbers) have been covered many times before in other general accounts, the slant that Stillwell takes on them is refreshing. This made the book satisfying to read and makes one ponder what other themes may underlay the development of mathematics (a train of thought that this reviewer considers worthwhile). Stillwell's theme added some coherence to all the different topics discussed in the book, without which the book would be less than what it is.

Stillwell does not try to avoid mathematical symbolism and argumentation, unlike many other authors of more general accounts of mathematics. He warns potential readers of this fact in the preface and encourages them to battle on with the material nonetheless, as “there is still no royal road to mathematics.” This book would be accessible to readers with a good high school education, but it would be most appropriate for those who enjoyed a course or two on mathematics at college or university, but because of life circumstances, were not able to continue their mathematical studies beyond that point. It is also fitting for those with more background in mathematics who would like, at the same time, to have a pleasant mathematical read and to gain a better understanding of the mathematical enterprise. Stillwell weaves historical details into his writing seamlessly, helping to give the reader the true feeling that mathematics is more than just a bunch of people playing games with symbols, but rather a rich and rewarding intellectual endeavor important to the human enterprise.

A good test of the quality of a book under review is to ask the following: if the reviewer was given the opportunity (either for herself or as a gift for someone else), would she buy the book? In this particular case, the reviewer answers yes.

Marcus Emmanuel Barnes is a graduate student studying the history of mathematics at Simon Fraser University in beautiful British Columbia, Canada. He can often be found passing the time in the company of books, especially those related to mathematics and science.

Comments

This is a wonderful book about the subject of mathematics. I learned a lot from this book which can be used in general class discussion to motivate students in a particular field of mathematics. It contains some real gems, such as:

The word surd comes from the Latin surdus meaning deaf.

The product of a sum of two squares is itself the sum of two squares.

A simple example of an elliptic function is the height of a point on an ellipse given in terms of the arc length.

The relationship between Gaussian primes and ordinary primes.

Hamilton's quaternions, which is another number system.

3-D symmetry is rare because only 5 regular polyhedra are fully symmetrical.

If 1 was prime then we would not have unique prime factorisation.

27 is the only cube which is 2 more than a square.

There are many others like this.

However this book is not really for the layman, because, as the author says in his preface, "many of the ideas are hard and there is no way to soften them'. The author has made extensive and fantastic use of illustrations to soften the blow and give an intuitive view of the mathematics, in particular in the chapter on curved space.

I really do like the historical context the author has placed his mathematics. I did not know that the Chinese had approximated π by the fraction 355/113, accurate to 6 decimal places. Nor did I know that the Leibniz series was known to Indian mathematician(s) well before it was discovered by Leibniz. (However, the author does not give the name(s) of the Indian mathematician(s) who had discovered this series.)

The chapter on the fourth dimension has an excellent historical context where the author describes Hamilton's search for the arithmetic of triples, quadruples etc. The chapter on the relationship between primes and Gaussian primes is also wonderful. The last chapter is a nice description of the different kinds of infinity and countable sets, including the interesting tidbit that "Many people believe the continuum hypothesis put Cantor in a mental hospital,'

In general there are some really good statements in the book which can be used to hook students to study mathematics, such as "Calculus as we know it today, is perhaps, the most powerful mathematical tool ever developed' and "Infinitesimal calculus brought almost the whole physical world within the scope of mathematics.'

I do have a number of minor quibbles. Sometimes a proposition or theorem is not clearly stated and it is only when you are half way through a proof you realise what is happening. There are statements which do not relate clearly to mathematical reality, such as "the square root of a half a turn,' which turns out to mean a rotation though a right angle.

There are a number of places where the book jumps from a simple introduction to the more advanced level within a few pages. For example, we are introduced to the idea of a complex number and within ten pages we are discussing Bézout's Theorem. In places I am not to sure what the author means. For example, on page 45 he says "complex analysis is known for its regularity and order.' Clearly complex numbers don't have order, so what does he mean?